   Chapter 8.4, Problem 6ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove that the distinct equivalence classes of the relation of congruence modulo n are the sets [ 0 ] ,   [ 1 ] ,   [ 2 ] ,   … ,   [ n − 1 ] , where for each a = 0 ,   1 ,   2 ,   … ,   n − 1 , [ a ] = { m ∈ Z | m ≡ a ( mod ​ n ) } .

To determine

To prove that the distinct equivalence classes of the relation of congruence modulo n are the sets ,,...[n1].

Explanation

Given information:

a={mZ|m=a(modn)} where for each a=0,1,2,....n1

Calculation:

Let us suppose m is the positive integer which is divided by n, so now using the Euclid’s division lemma, the following equation can be written −

m=nq+r;0r<n

So from the above equation the possible remainders when m is divided by n will be 0,1,2,.....n1.

Now let us construct the equivalence classes of the above remainders which can be ,,

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